Unitarily invariant norm inequalities involving $G_1$ operators

Publication date: 9 January 2018

Abstract: In this paper, we present some upper bounds for unitarily invariant norms inequalities. Among other inequalities, we show some upper bounds for the Hilbert-Schmidt norm. In particular, we prove �egin{align*} |f(A)Xg(B)pm g(B)Xf(A)|_2leq left|frac{(I+|A|)X(I+|B|)+(I+|B|)X(I+|A|)}{d_Ad_B} ight|_2, end{align*} where

A, B, X i n m a t h b b M_{n}

such that

A

,

B

are Hermitian with

s i g m a (A) c u p s i g m a (B) s u b s e t m a t h b b D

and

f, g

are analytic on the complex unit disk

m a t h b b D

,

g (0) = f (0) = 1

,

e x t r m R e (f) > 0

and

e x t r m R e (g) > 0

.

Mathematics Subject Classification ID

Operator theory (47-XX) Functional analysis (46-XX)

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